Showing papers on "Model order reduction published in 2003"

A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices

Abstract: In this paper, we present an approach to nonlinear model reduction based on representing a nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components as piecewise linear and then composing hundreds of components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a "training input." Computational results and performance data are presented for an example of a micromachined switch and selected nonlinear circuits. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or recently developed quadratic reduction techniques. Also, we propose a procedure for a posteriori estimation of the simulation error, which may be used to determine the accuracy of the extracted trajectory piecewise-linear reduced-order models. Finally, it is shown that the proposed model order reduction technique is computationally inexpensive, and that the models can be constructed "on the fly," to accelerate simulation of the system response.

Chaos in fractional-order autonomous nonlinear systems

Abstract: We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in ð0; 1� , based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic ‘‘jerk’’ model. In both models, numerical simulations are used to demonstrate that for different types of model nonlinearities, and using the proper control parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2 þ e ,1 > e > 0, using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by contraction and translation. Robustness against model order reduction is demonstrated. 2002 Elsevier Science Ltd. All rights reserved.

Guaranteed passive balancing transformations for model order reduction

Abstract: The major concerns in state-of-the-art model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms, such as PRIMA, generate guaranteed-passive models for systems with special internal structure, using numerically stable and efficient Krylov-subspace iterations. Truncated balanced realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper, we show how to construct TBR-like methods that generate guaranteed passive reduced models and in addition are applicable to state-space systems with arbitrary internal structure.

A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems

Abstract: In this study we discuss the problem of Model Order Reduction (MOR) for a class of nonlinear dynamical systems. In particular, we consider reduction schemes based on projection of the original state-space to a lower-dimensional space e.g. by using Krylov methods. In the nonlinear case, however, applying a projection-based MOR scheme does not immediately yield computationally efficient macromodels. In order to overcome this fundamental problem, we propose to first approximate the original nonlinear system with a weighted combination of a small set of linearized models of this system, and then reduce each of the models with an appropriate projection method. The linearized models are generated about a state trajectory of the nonlinear system corresponding to a certain ‘training’ input. As demonstrated by results of numerical tests, the obtained trajectory quasi-piecewise-linear reduced order models are very cost-efficient, while providing superior accuracy as compared to existing MOR schemes, based on single-state Taylor’s expansions. In this dissertation, the proposed MOR approach is tested for a number of examples of nonlinear dynamical systems, including micromachined devices, analog circuits (discrete transmission line models, operational amplifiers), and fluid flow problems. The tests validate the extracted models and indicate that the proposed approach can be effectively used to obtain system-level models for strongly nonlinear devices. This dissertation also shows an inexpensive method of generating trajectory piecewise-linear (TPWL) models based on constructing the reduced models ‘on-the-fly’, which accelerates simulation of the system response. Moreover, we propose a procedure for estimating simulation errors, which can be used to determine accuracy of the extracted trajectory piecewise-linear reduced order models. Finally, we present projection schemes which result in improved accuracy of the reduced order TPWL models, as well as discuss approaches leading to guaranteed stable and passive TPWL reduced-order models. Thesis Supervisor: Jacob K. White Title: Professor of Electrical Engineering and Computer Science

Piecewise polynomial nonlinear model reduction

Abstract: We present a novel, general approach towards model-order reduction (MOR) of nonlinear systems that combines good global and local approximation properties. The nonlinear system is first approximated as piecewise polynomials over a number of regions, following which each region is reduced via polynomial model-reduction methods. Our approach, dubbed PWP, generalizes recent piecewise linear approaches and ties them with polynomial-based MOR, thereby combining their advantages. In particular, reduced models obtained by our approach reproduce small-signal distortion and intermodulation properties well, while at the same time retaining fidelity in large-swing and large-signal analyses, e.g., transient simulations. Thus our reduced models can be used as drop-in replacements for time-domain as well as frequency-domain simulations, with small or large excitations. By exploiting sparsity in system polynomial coefficients, we are able to make the polynomial reduction procedure linear in the size of the original system. We provide implementation details and illustrate PWP with an example.

A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and MEMS

Abstract: In this paper we propose a method for generating reduced models for a class of nonlinear dynamical systems, based on truncated balanced realization (TBR) algorithm and a recently developed trajectory piecewise-linear (TPWL) model order reduction approach. We also present a scheme which uses both Krylov-based and TBR-based projections. Computational results, obtained for examples of nonlinear circuits and a micro-electro-mechanical system (MEMS), indicate that the proposed reduction scheme generates nonlinear macromodels with superior accuracy as compared to reduction algorithms based solely on Krylov subspace projections, while maintaining a relatively low model extraction cost.

Well-conditioned asymptotic waveform evaluation for finite elements

Abstract: The frequency-domain finite-element method (FEM) results in matrix equations that have polynomial dependence on the frequency of excitation. For a wide-band fast frequency sweep technique based on a moment-matching model order reduction (MORe) process, researchers generally take one of two approaches. The first is to linearize the polynomial dependence (which will either limit the bandwidth of accuracy or require the introduction of extra degrees of freedom) and then use a well-conditioned Krylov subspace technique. The second approach is to work directly with the polynomial matrix equation and use one of the available, but ill-conditioned, asymptotic waveform evaluation (AWE) methods. For large-scale FEM simulations, introducing extra degrees of freedom, and therefore increasing the length of the MORe vectors and the amount of memory required, is not desirable; therefore, the first approach is not alluring. On the other hand, an ill-conditioned AWE process is unattractive. This paper presents a novel MORe technique for polynomial matrix equations that circumvents these problematic issues. First, this novel process does not require any additional unknowns. Second, this process is well-conditioned. Along with the presentation of the novel algorithm, which is called well-conditioned AWE (WCAWE), numerical examples modeled using the FEM are given to illustrate its accuracy.

NORM: compact model order reduction of weakly nonlinear systems

Abstract: This paper presents a compact Nonlinear model Order Reduction Method (NORM) that is applicable for time-invariant and time-varying weakly nonlinear systems. NORM is suitable for reducing a class of weakly nonlinear systems that can be well characterized by low order Volterra functional series. Unlike existing projection based reduction methods by J. Roychowdhury et al. (1999), NORM begins with the general matrix-form Volterra nonlinear transfer functions to derive a set of minimum Krylov subspaces for order reduction. Direct moment matching of the nonlinear transfer functions by projection of the original system onto this set of minimum Krylov subspaces leads to a significant reduction of model size. As we will demonstrate as part of our comparison with existing methods, the efficacy of model order for weakly nonlinear systems is determined by the extent to which models can be reduced. Our results further indicate that a multiple-point version of NORM can substantially reduce the model size and approach the ultimate model compactness that is achievable for nonlinear system reduction. We demonstrate the practical utility of NORM for macromodeling weakly nonlinear RF circuits with time-varying behavior.

Reduced order models of refined Yee's cells

Abstract: This letter introduces a new approach to increasing the accuracy of Finite Difference (FD) methods by means of local mesh refinement. The area slightly larger than single Yee's cell is covered by dense mesh and its macromodel is created by the Model Order Reduction (MOR) of state equations in the frequency domain. Such macromodels are subsequently used in the Finite Difference Time Domain (FDTD) or the Finite Difference Frequency Domain (FDFD) analysis of the entire structure. Unlike a popular subgridding technique, the model order reduction approach does not affect the stability or convergence properties of underlying numerical schemes.

Broadband model order reduction of polynomial matrix equations using single-point well-conditioned asymptotic waveform evaluation: derivations and theory

Abstract: SUMMARY To eect a model order reduction (MORe) process on a system which has a polynomial matrix equation dependence on the MORe parameter, researchers generally take one of two approaches. Therst is to linearize the system by introducing extra degrees of freedom and then to solve the resulting expanded, linear system with a method such as Lanczos or Arnoldi. The second approach is to work directly with the polynomial system and use a technique such as asymptotic waveform evaluation (AWE). Of course, each approach has advantages and disadvantages. In this paper, a new technique will be presented which has some desirable characteristics from both approaches and which is able to circumvent simultaneously some of their disadvantages. It can be shown that both the Arnoldi and the AWE methods are special cases of this new technique. Finally, numerical results will show the viability of the new method, which will be called the well-conditioned asymptotic waveform evaluation (WCAWE) method. Copyright ? 2003 John Wiley & Sons, Ltd.

Model-order reduction of nonlinear models of electromagnetic phased-array hyperthermia

Abstract: A method based on the Karhunen-Loeve (KL) transform is proposed for the reduction of large-scale, nonlinear ordinary differential equations such as those arising from the finite difference modeling of biological heat transfer. The method of snapshots is used to expedite computation of the required quantities in the KL procedure. Guidelines are presented and validated for snapshot selection and resultant basis series truncation, emphasizing the special physical features of the electromagnetic phased-array heat transfer physics. Applications to fast temperature prediction are presented.

Realizable RLCK circuit crunching

Abstract: Reduction of an extracted netlist is an important pre-processing step for techniques such as model order reduction in the design and analysis of VLSI circuits. This paper describes a method for realizable reduction of RLCK netlists by node elimination. The method is much faster than model order reduction technique and hence is appropriate as a pre-processing step. The proposed method eliminates nodes with time constants below a user specified time constants. By giving the freedom to the user to select a critical point in the spectrum of nodal time constants, this method provides an option to make a tradeoff between accuracy and reduction. The proposed method preserves the dc characteristics and the first two moments at all nodes. It also recognizes and eliminates all the redundant inductances generated by the extraction tools. The proposed method naturally reduces to TICER according to B. N. Sheehan (1999) in the absence of any inductances.

Extraction of heat transfer macromodels for MEMS devices

Abstract: A model order reduction (MOR) technique for heat-transfer system-level modeling is presented. A heat transfer solver using the finite difference method (FDM), which is appropriate for MEMS thermal analysis, is implemented. The numerical models generated by the FDM solver then were successfully reduced into low-order compact models by an Arnoldi-based model-order-reduction technique. The compact macromodels can be easily inserted into system-level or circuit simulators. We also demonstrated that the macromodel results for a thermal actuator are in good agreement with the experimental results.

RCS computation over a frequency band using the characteristic basis and model order reduction method

Abstract: In many practical applications, it is desirable to analyze the radar scattering from an object over a wide frequency band. Earlier attempts to achieving a fast frequency sweep involved the computation of higher order moments of the MoM matrix, which requires considerable fill time for each frequency point when the number of unknowns is large. In this paper, we present an approach for constructing a set of universal basis functions that can be used over the entire frequency band of interest. The use of this universal basis set obviates the need to factorize the block sub-matrices at each frequency, and results in considerable time saving. The numerical accuracy and the computational advantage of the proposed technique are illustrated by studying the problem of plane wave scattering from a PEC plate over a wide frequency band.

Improved model-order reduction by using spacial information in moments

Abstract: The new concept of multinode moment matching (MMM) is introduced in this paper. The MMM technique simultaneously matches the moments at several nodes of a circuit using explicit moment matching around s=0. As compared to the well known single-point moment matching (SMM) techniques (such as asymptotic waveform evaluation), MMM has several advantages. First, the number of moments required by MMM is significantly lower than SMM for a reduced-order model of the same accuracy, which directly translates into computational efficiency. This higher computational efficiency of MMM as compared to SMM increases with the number of inputs to the circuit. Second, MMM has much better numerical stability as compared to SMM. This characteristic allows MMM to calculate an arbitrarily high-order approximation of a linear system, achieving the required accuracy for systems with complex responses. Finally, MMM is highly suitable for parallel-processing techniques especially for higher order approximations while SMM has to calculate the moments sequentially and cannot be adapted to parallel processing techniques.

Model order reduction of nonuniform transmission lines using integrated congruence transform

Abstract: This paper presents a new algorithm based on integrated congruence transform for the analysis of both uniform and nonuniform transmission lines. The key advantage of the proposed algorithm is that constructing a spanning orthonormal basis for the space-dependent moments is done without computing these moments explicitly. The proposed algorithm thus carries the numerical efficiency of Krylov-based projection techniques of lumped RLC networks to the domain of the distributed transmission line networks. The proposed algorithm can be used to construct an orthogonal basis for any set of moments related through a differential operator.

SuPREME: Substrate and Power-delivery Reluctance-Enhanced Macromodel Evaluation

Abstract: The recent demand for system-on-chip RF mixed-signal design and aggressive supply-voltage reduction require chip-level accurate analysis of both the substrate and power delivery systems. Together with the rising frequency, low-k dielectric, copper interconnects, and high conductivity substrate, the inductance effects raised serious concern recently. However, the increasing design complexity creates tremendous challenges for chip-level power-delivery substrate co-analysis. In this paper, we propose a novel and efficient reluctance-based passive model order reduction technique to serve these tasks. Our work, SuPREME(Substrate and Power-delivery Reluctance-Enhanced Macromodel Evaluation) not only greatly reduces the computational complexity of previous reluctance-based model order algorithms but is also capable of handling large number of noise sources efficiently. To facilitate the analysis of inductive substrate return paths and evaluate the high-frequency substrate coupling effects, we derive a novel RLKC substrate model from Maxwell's equations for the first time. Experimental results demonstrate the superior runtime and accuracy of SuPREME compared to the traditional MNA-based simulation.

Speech modelling by model-order reduction: SNR behaviour

Abstract: Using model reduction, a new approach for low-order speech modelling is presented. The process starts with a relatively high-order (full-order) autoregressive (AR) model obtained by some classical methods. The AR model is then reduced using the state projection method. The model reduction yields a reduced-order autoregressive moving-average model which interestingly preserves the key properties of the original model such as stability, minimality, and phase minimality. SNR behaviour are investigated. To illustrate the performance and the effectiveness of the proposed approach, computer simulations are conducted on practical speech segments.

Model-Order Reduction Based on PRONY's Method

Abstract: A new model-order reduction technique for linear dynamic systems is presented. The idea behind this technique is to transform the dynamic system function from the s-domain into the z-domain via the bilinear transformation, then use Pronyýs [1] least-squares approximation method instead of the commonly employed Pade approximation method, and finally transform the reduced system back into the s-domain using the inverse bilinear transformation. Simulation results for large practical systems show that this technique based on Pronyýs method gives much higher accuracy than the traditional Pade method, and results in lower-order approximations without significant increase in simulation time.

Structural Dynamic Frequency Response Using Combined Direct and Adjoint Reduced-Order Approximations

Abstract: A second-order combined approximate-direct/approximate-adjoint method for efficient reduced-order calculation of frequency response in structural dynamics is presented. The method is design oriented. It focuses not only on the response itself (in the form of displacements and, especially, stresses) but also on sensitivities of the response with respect to structural design variables. Comparisons of approximate results obtained by mode-displacement order reduction, reduced-order adjoint solutions based on Ritz vectors, and the second-order approximate method demonstrate the accuracy of the second-order method, and provide lessons that suggest directions for future research.

A fast filter optimization scheme based on model order reduction

Abstract: This paper presents a novel, fast optimization technique that is especially suitable for the design of microwave filters. For the discretization of the structure, the well established Finite Integration Technique (FIT) is used. A recently published two step Model Order Reduction (MORe) technique computes the poles and residuals of the structure's impedance matrix as well as its sensitivity to the geometry parameters. The actual optimization is then performed in the reduced order space of the poles and residuals. This significantly reduces the number of full-wave simulation runs, that is usually associated with the optimization of filters.

A novel AGC simulation scheme based on reduced order prime mover models

Abstract: This paper presents a new simulation scheme for AGC studies of power system. The proposed scheme uses a different set of assumptions from that conventionally used. In this scheme, all the areas have been considered to be operating at the same frequency. Further, the proposed approach can preserve the identity of each generating unit. Additionally, the computational complexity has been reduced by resorting to lower order generating unit models. In this context, model order reduction techniques have been used to obtain lower order models for AGC studies. The effectiveness of the new simulation approach is demonstrated by considering the IEEE 30 bus test system as a 3 area system.

PODEA: Power delivery efficient analysis with realizable model reduction

Abstract: The huge number of independent sources of power delivery systems prevents the use of traditional model reduction algorithms due to the port domination nature. This paper presents an innovative RC model reduction method, PODEA, to analyze RC linear circuits with many dynamic independent sources. Based on the multi-port Norton theorem and model order reduction techniques, we develop and apply current source transformation algorithms to transform attached current sources from one node to neighboring nodes. Since there is no source attached, general RC reduction algorithms can be applied to eliminate the node. Experimental results demonstrate the efficiency and accuracy of the proposed PODEA algorithm. With linear running time, for an example with over 50,000 nodes, our reduction method only takes about 0.6 seconds while maintaining 1% error and 88% reduction ratio.

Optimal Model Order Reduction for Maximal Real Part Norms

Abstract: An observation is made that a polynomial time algorithm exists for the problem of optimal model order reduction of finite order systems in the case when the approximation error measure to be minimized is defined as maximum of the real part of the model mismatch transfer function over a certain set of frequencies. Applications to H-infinity model reduction and comparison to the classical Hankel model reduction are discussed.

Symbolic model order reduction

Abstract: Symbolic model order reduction (SMOR) is the problem of reducing a large circuit that contains symbolic circuit parameters to smaller low order models at its ports. Several methods, including symbol isolation, single frequency point reduction, and multiple frequency point reduction, are described and compared. Test circuits with simulation results are presented to demonstrate the accuracy and efficiency of SMOR.

Model order reduction of electromagnetic systems and RLC circuits for power plane resonance analysis

Abstract: This paper describes an efficient method for generating simulation macromodels of power plane resonances on printed circuit boards (PCB). This method models a PCB as a hybrid system of equations which is composed of electromagnetic systems and RLC circuits. The model order reduction technique is utilized in order to construct macromodels from the hybrid system of equations.

Application of model order reduction techniques to compact FDFD method for guided wave structures

Abstract: A compact FDFD method in conjuncture with model order reduction (MOR) techniques has been investigated for guided wave structures. The results of this study show that these MOR techniques can further improve the computational efficiency of the compact FDFD method. Numerical examples demonstrate the accuracy of these techniques.